The Krull–Schmidt Theorem

Q: Does Krull–Schmidt need both chain conditions?

Existence needs only one: the ACC or the DCC alone suffices for (19.20). Uniqueness as packaged in (19.22) needs finite length, i.e. both — because it is proved by way of (19.17), and that result genuinely requires both. What uniqueness really needs is local endomorphism rings, which finite length happens to guarantee.

Q: Why must the summands be strongly indecomposable and not merely indecomposable?

The proof writes id_M_1 as a sum of s endomorphisms of M_1 and needs one of them to be invertible. That inference is exactly the defining property (19.1)(5') of a local ring. An indecomposable module only guarantees that End(M_1) has no nontrivial idempotents, which does not force any summand of a sum to be a unit.

Q: Where does uniqueness first fail?

Over any Dedekind domain with nontrivial class group. If [A] has order 2 then A ⊕ A ≅ R ⊕ A^2 ≅ R ⊕ R, and A ≅ R. Swan's example pushes the failure into commutative noetherian local domains, so being local is no protection.

Q: Is Krull–Schmidt the same as Jordan–Hölder?

No. Jordan–Hölder is about composition factors of a filtration and holds for every module of finite length over every ring. Krull–Schmidt is about indecomposable direct summands. A module can have a unique list of composition factors while its decomposition into indecomposables is not unique, and the two lists coincide only when the module is semisimple.

Q: Does the theorem extend to infinite direct sums?

There is a genuine infinite version — modules with local endomorphism rings satisfy strong exchange properties, and the Crawley–Jónsson–Warfield theory develops this — but it is a separate theorem. The finite induction in the proof above does not extend on its own.

Q: What does the theorem give me computationally?

It makes the output of a decomposition algorithm an invariant. Once you know a module over a finite-dimensional algebra decomposes as 2U_1 ⊕ 3U_2, no other decomposition into indecomposables exists, so the pair of multiplicities is a valid fingerprint for isomorphism testing.

Executive Summary

The Krull–Schmidt theorem is two statements bolted together, and they have different price tags. Existence — every module with the ACC or the DCC on submodules is a finite direct sum of indecomposables — is elementary and costs one chain condition $(19.20)$ . Uniqueness — the number of summands and their isomorphism types are determined up to permutation — is the hard half and is false in general.

What buys uniqueness is Azumaya's hypothesis: the summands of one of the two decompositions must have local endomorphism rings $(19.21)$ . Over a module of finite length that hypothesis is automatic, because an indecomposable module of finite length has local endomorphism ring $(19.17)$ . That is precisely why the classical Krull–Schmidt theorem $(19.22)$ holds for finite length and evaporates one step outside it.

ACC or DCCExistence

Local $End$ Uniqueness

1950Azumaya's version

$ℤ [\sqrt{- 5}]$ Standard failure

Overview

Decomposition theorems are the standard way to reduce a classification problem to a finite list of atoms. For modules the atoms are the indecomposables, and the theorem one wants is that the atoms and their multiplicities are invariants of the module.

M = M_{1} \oplus \dots \oplus M_{r} = N_{1} \oplus \dots \oplus N_{s} ⟹ r = s and M_{i} ≅ N_{π (i)}

(19.21)

The conclusion of Krull–Schmidt–Azumaya, for some permutation $π$ . The hypothesis is that every $N_{j}$ is indecomposable and every $M_{i}$ is strongly indecomposable.

Note the asymmetry in the hypothesis: only one family has to be strongly indecomposable. Since the conclusion identifies the two families up to isomorphism, the $N_{j}$ turn out to be strongly indecomposable after the fact — but they are not assumed to be, and that is what makes the theorem usable.

This page treats existence $(19.20)$ , the Azumaya uniqueness theorem $(19.21)$ and its two working corollaries: modules of finite length $(19.22)$ and finitely generated modules over a one-sided artinian ring $(19.23)$ . The failure of uniqueness over Dedekind domains, and Swan's sharper failure over a commutative noetherian local domain, mark the boundary.

Learning Objectives

Prove $(19.20)$ : ACC or DCC alone forces a finite decomposition into indecomposables.
State $(19.21)$ with its exact one-sided hypothesis and reproduce Azumaya's proof.
Locate the step in that proof where $(19.1) (5')$ is used, and explain why nothing weaker suffices.
Deduce $(19.22)$ and $(19.23)$ and say which chain conditions each needs.
Exhibit $𝔄 \oplus 𝔄 ≅ R \oplus R$ over $ℤ [\sqrt{- 5}]$ and identify which hypothesis fails.
Use uniqueness to prove cancellation: $t \cdot M ≅ t \cdot N$ implies $M ≅ N$ over a right artinian ring.

Definitions

Krull–Schmidt decomposition: A finite direct sum decomposition $M = M_{1} \oplus \dots \oplus M_{r}$ with each $M_{i}$ indecomposable. The zero module is the empty sum.
Indecomposable: $M \neq 0$ with no splitting $M = A \oplus B$ , $A, B \neq 0$ ; equivalently $End (M)$ has only the idempotents $0$ and $1$ .
Strongly indecomposable: $M \neq 0$ with $End (M)$ a local ring $(19.12)$ .
Finite length: $M$ has a composition series; equivalently $M$ satisfies both the ACC and the DCC on submodules.
$t \cdot M$: The direct sum of $t$ copies of $M$ .
Split monomorphism: An injection $u : A \to B$ whose image is a direct summand of $B$ ; equivalently $v u = {id}_{A}$ for some $v$ .

Modules here are right modules; every statement has a left analogue obtained by working over the opposite ring.

Core Concepts

Existence is cheap

Call a submodule good if it has a Krull–Schmidt decomposition. The zero module is good, every indecomposable submodule is good, and a direct sum of two good submodules is good. If $M$ itself were bad, then it is not indecomposable, so $M = M_{1} \oplus M_{1}'$ with both factors nonzero and at least one — say $M_{1}'$ — bad. Iterating produces a strictly descending chain of bad submodules and a strictly ascending chain of good sums, so $M$ satisfies neither chain condition.

ACC or DCCfinite decomposition existssummands are indecomposable

Uniqueness is expensive

Given two decompositions, write $1 = α_{1} + \dots + α_{r} = β_{1} + \dots + β_{s}$ in $E = End (M)$ for the two families of projections. Restricting $α_{1} = α_{1} β_{1} + \dots + α_{1} β_{s}$ to $M_{1}$ writes the identity of $End (M_{1})$ as a sum of $s$ endomorphisms. If $End (M_{1})$ is local, $(19.1) (5')$ says one of those summands is a unit — that is, an automorphism of $M_{1}$ . Everything else is bookkeeping.

Why finite length is the right sufficient condition

For a module of finite length the two halves combine cleanly. Finite length gives both chain conditions, so $(19.20)$ produces a decomposition; and each indecomposable summand again has finite length, so $(19.17)$ makes its endomorphism ring local. Both hypotheses of $(19.21)$ are then met by both decompositions, and uniqueness follows.

Key Results

Proposition(19.20)Existence of a Krull–Schmidt decomposition

Let $R$ be any ring and $M_{R}$ a right $R$ -module whose submodules satisfy either the ascending or the descending chain condition. Then $M$ is a finite direct sum of indecomposable submodules.

Proof

Say a submodule is good if it decomposes as a finite direct sum of indecomposables, and bad otherwise. The zero module is good (empty sum), any indecomposable submodule is good, and if $N, N'$ are good with $N \cap N' = 0$ then $N \oplus N'$ is good.

Suppose $M$ is bad. Then $M$ is not indecomposable and not zero, so $M = M_{1} \oplus M_{1}'$ with $M_{1}, M_{1}' \neq 0$ . If both were good, $M$ would be good; so one is bad, say $M_{1}'$ . Repeating inside $M_{1}'$ gives $M_{1}' = M_{2} \oplus M_{2}'$ with $M_{2}, M_{2}' \neq 0$ and $M_{2}'$ bad, and so on.

This produces $M ⊋ M_{1}' ⊋ M_{2}' ⊋ \dots$ , violating the DCC, and simultaneously $0 ⊊ M_{1} ⊊ M_{1} \oplus M_{2} ⊊ M_{1} \oplus M_{2} \oplus M_{3} ⊊ \dots$ , violating the ACC. Either chain condition therefore rules out a bad $M$ .

Theorem(19.21)Krull–Schmidt–Azumaya

Let $R$ be a ring and let a right $R$ -module $M$ have two decompositions into submodules

M = M_{1} \oplus \dots \oplus M_{r} = N_{1} \oplus \dots \oplus N_{s},

where every $N_{j}$ is indecomposable and every $M_{i}$ is strongly indecomposable, i.e. $End (M_{i})$ is a local ring. Then $r = s$ , and after reindexing $M_{i} ≅ N_{i}$ for $1 \leq i \leq r$ .

Proof

Induct on $r$ . Let $α_{i} : M \to M_{i} \subseteq M$ and $β_{j} : M \to N_{j} \subseteq M$ be the projections attached to the two decompositions, viewed in $E = End (M_{R})$ , so that $1 = α_{1} + \dots + α_{r} = β_{1} + \dots + β_{s}$ .

Step 1: find a matching summand. Each $α_{1} β_{j}$ maps $M$ into $M_{1}$ , so restriction to $M_{1}$ gives elements of $End (M_{1})$ , and $\sum_{j = 1}^{s} (α_{1} β_{j}) |_{M_{1}} = α_{1} |_{M_{1}} = {id}_{M_{1}}$ . Since $End (M_{1})$ is local, condition $(19.1) (5')$ gives some $j$ with $(α_{1} β_{j}) |_{M_{1}}$ an automorphism of $M_{1}$ . Reindex so that $j = 1$ .

Step 2: upgrade to an isomorphism. Write $θ = (α_{1} |_{N_{1}}) \circ (β_{1} |_{M_{1}}) \in Aut (M_{1})$ . Then $θ^{- 1} (α_{1} |_{N_{1}})$ is a left inverse for $β_{1} |_{M_{1}} : M_{1} \to N_{1}$ , so $β_{1} |_{M_{1}}$ is a split monomorphism and its image is a nonzero direct summand of $N_{1}$ . As $N_{1}$ is indecomposable, that image is all of $N_{1}$ ; hence $β_{1} |_{M_{1}} : M_{1} \to N_{1}$ is an isomorphism.

Step 3: exchange. We claim $M = M_{1} \oplus N_{2} \oplus \dots \oplus N_{s}$ . Since $β_{1} |_{M_{1}}$ is injective, $M_{1} \cap \ker (β_{1}) = 0$ , and $\ker (β_{1}) = N_{2} \oplus \dots \oplus N_{s}$ . For the sum: given $a \in N_{1}$ , surjectivity in Step 2 gives $b \in M_{1}$ with $β_{1} (b) = a$ ; then $β_{1} (a - b) = a - a = 0$ , so $a - b \in N_{2} \oplus \dots \oplus N_{s}$ and $a \in M_{1} + N_{2} + \dots + N_{s}$ . Hence $M = N_{1} + \dots + N_{s} \subseteq M_{1} + N_{2} + \dots + N_{s}$ , proving the claim.

Step 4: induct. Quotienting the two descriptions of $M$ by $M_{1}$ gives $N_{2} \oplus \dots \oplus N_{s} ≅ M / M_{1} ≅ M_{2} \oplus \dots \oplus M_{r}$ , a module with one fewer strongly indecomposable summand. The case $r = 1$ is immediate: $M = M_{1}$ is indecomposable, so $s = 1$ . Applying the inductive hypothesis finishes the proof.

Corollary(19.22)Krull–Schmidt Theorem

Let $M_{R}$ be a right $R$ -module of finite composition length over an arbitrary ring $R$ . Then $M = M_{1} \oplus \dots \oplus M_{r}$ with each $M_{i}$ an indecomposable submodule. Moreover $r$ is uniquely determined, and the sequence of isomorphism types $M_{1}, \dots, M_{r}$ is uniquely determined up to a permutation.

Proof

Finite length gives both chain conditions, so $(19.20)$ provides the decomposition. Each $M_{i}$ is a submodule of $M$ , hence of finite length, and indecomposable, so $End (M_{i})$ is local by $(19.17)$ ; every $M_{i}$ is therefore strongly indecomposable. Given a second decomposition into indecomposables, $(19.21)$ applies and yields $r = s$ together with the matching of isomorphism types.

Corollary(19.23)The artinian case

Both conclusions of $(19.22)$ hold for every finitely generated right module $M$ over a right artinian ring $R$ — in particular for every finitely generated module over a finite-dimensional algebra over a field.

Proof

Over a right artinian ring, a finitely generated right module has a composition series $(4.15)$ , so $(19.22)$ applies verbatim.

Corollary—Cancellation

Let $R$ be right artinian and $M, N$ finitely generated right $R$ -modules. If $t \cdot M ≅ t \cdot N$ for some integer $t \geq 1$ , then $M ≅ N$ .

Proof

By $(19.23)$ each of $M, N$ has a Krull–Schmidt decomposition with well-defined multiplicities. The multiplicity of an indecomposable type $U$ in $t \cdot M$ is $t$ times its multiplicity in $M$ , and likewise for $N$ . Equality of the two multisets therefore gives equality of all multiplicities after dividing by $t$ , and $M ≅ N$ .

Theorem(19.25)Noether–Deuring

Let $R$ be a finite-dimensional algebra over a field $k$ , let $M, N$ be right $R$ -modules of finite $k$ -dimension, and let $K \supseteq k$ be any field extension. If $M^{K} ≅ N^{K}$ as $R^{K} = R \otimes_{k} K$ -modules, then $M ≅ N$ as $R$ -modules.

The proof splits on the size of $k$ . When $| k | > \dim_{k} M$ , a nonvanishing-determinant polynomial argument produces an $R$ -isomorphism over $k$ directly. When $k$ is too small, one first passes to a finite extension $L$ and then uses the cancellation corollary above, since $M \otimes_{k} L ≅ t \cdot M$ as $R$ -modules with $t = [L : k]$ .

Proof Techniques and Method

How these proofs work, and which move to reuse.

Move 1

Turn decompositions into idempotents

Replace $M = ⨁ M_{i}$ by the orthogonal idempotents $α_{i} \in End (M)$ with $\sum α_{i} = 1$ . Comparing two decompositions becomes comparing two orthogonal resolutions of $1$ .

Move 2

Make a unit appear in a sum

A sum of endomorphisms equal to $id$ must contain a unit if the ring is local. This is the whole use of the hypothesis, and it is the reason local rather than indecomposable is the right condition.

Move 3

Split mono into indecomposable is iso

A split monomorphism with indecomposable target and nonzero source is an isomorphism, because its image is a nonzero direct summand. This converts a one-sided invertibility statement into a genuine matching of summands.

The exchange in Step 3 of the proof is the germ of a much larger theory. Modules for which such exchanges are always possible are said to have the exchange property, and Azumaya's theorem is the statement that modules with local endomorphism rings have it. The failures collected below are all failures of exchange.

Worked Example

Two genuinely different decompositions of $R^{2}$

Take $R = ℤ [\sqrt{- 5}]$ , the ring of algebraic integers of $ℚ (\sqrt{- 5})$ . It is a Dedekind domain of class number $2$ . Let

𝔄 = (3, 1 + \sqrt{- 5}) \subseteq R, N (𝔄) = 3 .

(E.1)

$𝔄$ is not principal: a generator would have norm $3$ , and $a^{2} + 5 b^{2} = 3$ has no solution in integers. A direct computation gives $𝔄^{2} = (\sqrt{- 5} - 2)$ , which is principal — for instance $- 4 + 2 \sqrt{- 5} = 2 (\sqrt{- 5} - 2)$ and $3 + 3 \sqrt{- 5} = (1 - \sqrt{- 5}) (\sqrt{- 5} - 2)$ , and the norms match at $9$ . So $[𝔄]$ has order $2$ in the class group.

For a Dedekind domain the Steinitz isomorphism $𝔄 \oplus 𝔅 ≅ R \oplus 𝔄 𝔅$ holds for all nonzero ideals. Applying it with $𝔅 = 𝔄$ :

𝔄 \oplus 𝔄 ≅ R \oplus 𝔄^{2} ≅ R \oplus R .

(E.2)

Two decompositions of the same module into indecomposables, with non-isomorphic summands.

Every nonzero ideal $I$ of a commutative domain is indecomposable as a module: if $I = A \oplus B$ with $0 \neq a \in A$ and $0 \neq b \in B$ , then $a b$ lies in both $A$ and $B$ and is nonzero, contradicting $A \cap B = 0$ . So $R$ and $𝔄$ are both indecomposable, and $𝔄 not ≅ R$ because $𝔄$ is not principal. Uniqueness fails.

A case where luck substitutes for the theorem

For $R = ℤ$ and $M$ a finitely generated abelian group, $M$ satisfies the ACC and decomposes into copies of $ℤ$ and of $ℤ / p^{n} ℤ$ , with uniqueness supplied by the fundamental theorem of finitely generated abelian groups. But $End (ℤ) ≅ ℤ$ is not local, so $(19.21)$ does not apply to the free summands. The uniqueness here is a fact about $ℤ$ , not an instance of Krull–Schmidt — and the Dedekind example above shows the argument does not survive the move from $ℤ$ to a general Dedekind domain.

The torsion summands do satisfy the hypothesis: $End (ℤ / p^{n} ℤ) ≅ ℤ / p^{n} ℤ$ is local, and $ℤ / p^{n} ℤ$ has finite length $n$ . So the torsion part of the classification is an instance of $(19.22)$ .

Process and Workflow

Can I use Krull–Schmidt uniqueness on my module $M$ ?

$M$ has finite lengthYes —

(19.22)

applies unconditionally, and every summand automatically has a local, indeed completely primary, endomorphism ring.

$M$ is f.g. over a one-sided artinian ringYes —

(19.23)

; the ring's chain condition supplies a composition series.

$M$ is noetherian but of infinite lengthOnly if you can verify that the summands of one decomposition have local endomorphism rings. Otherwise expect failure: the Dedekind example lives here.

$M$ has neither chain conditionEven existence is unavailable. Look for a different invariant — for infinite direct sums, Kaplansky-type or Crawley–Jónsson theory replaces Krull–Schmidt.

Confirm a chain conditionACC or DCC on submodules of

M

. Either one alone gives existence by

(19.20)

DecomposeSplit off indecomposable summands; over a finite-dimensional algebra this is what the Meataxe and related algorithms do.

Check endomorphism ringsFor each summand, decide whether

End (M_{i})

is local. Finite length makes this automatic via

(19.17)

Apply uniqueness and record multiplicitiesThe multiset of isomorphism types is now an invariant of

M

and may be used to distinguish modules and to cancel common summands.

Comparison and Classification

Where each half of Krull–Schmidt holds
Setting	Existence	Uniqueness	Reason
$M$ of finite length, any $R$	yes	yes	(19.22) via (19.17)
$M$ f.g. over right artinian $R$	yes	yes	(19.23) via (4.15)
$M$ f.g. over a finite-dimensional $k$ -algebra	yes	yes	special case of (19.23)
$M$ noetherian, summands with local $End$	yes	yes	(19.21) directly
$M$ f.g. over a PID	yes	yes	structure theorem, not (19.21)
$M$ f.g. over a Dedekind domain	yes	no	Steinitz; class group obstruction
$M$ f.g. over a commutative noetherian local domain	yes	no	Swan's example
$M$ with neither chain condition	no	n/a	(19.20) does not apply

Hypotheses on the summands, and what each yields
	Existence	Azumaya applies	Multiplicities unique
Indecomposable only	yes	no	no
$End$ a domain	yes	no	no
$End$ local	yes	yes	yes
Finite length	yes	yes	yes
Simple	yes	yes	yes

Hypotheses on the summands, and what each yields

Relationship Map

The logical dependencies are strictly layered: the classical statements are corollaries of Azumaya's, which is a corollary of nothing.

(19.21) Krull–Schmidt–Azumaya — uniqueness from local endomorphism rings
- with (19.20) and (19.17)
  - (19.22) Krull–Schmidt for finite length
  - (19.23) f.g. modules over right artinian rings
  - cancellation: $t \cdot M ≅ t \cdot N \Rightarrow M ≅ N$
- feeds into
  - (19.25) Noether–Deuring theorem
  - (19.30) Dickson's divisibility theorem
  - principal indecomposable modules and blocks
- fails without local End
  - $𝔄 \oplus 𝔄 ≅ R \oplus R$ over $ℤ [\sqrt{- 5}]$
  - Swan's local-domain example
  - restored by completion — see §21

M

finite length

M_{i}

indecomposable

End (M_{i})

localuniqueness

Applications and Industry Use

Applications here means where this structure is used — inside mathematics and in the engineering and computing disciplines that consume it.

Representation theory

Principal indecomposables and blocks

Over a right artinian ring, $R_{R} = U_{1} \oplus \dots \oplus U_{n}$ is a Krull–Schmidt decomposition; the $U_{i}$ are the principal indecomposable modules and their invariance underlies the whole theory of blocks and Cartan matrices in modular representation theory.

Field extensions

Noether–Deuring

Uniqueness is what lets you decide isomorphism of representations after enlarging the field: $M^{K} ≅ N^{K}$ forces $M ≅ N$ $(19.25)$ . Computationally this means one may work over a convenient splitting field and descend.

Computer algebra

Canonical form of a module

GAP and Magma return module decompositions as a list of indecomposables with multiplicities. That output is only well defined because of $(19.23)$ ; over rings where uniqueness fails, such a list is not an invariant.

Commutative algebra

Vector bundles and projective modules

The Steinitz classification of f.g. projectives over a Dedekind domain is exactly the measured failure of Krull–Schmidt: rank plus a class-group invariant, rather than a multiset of atoms.

The theorem is also a piece of infrastructure for category theory: Atiyah transported it to coherent sheaves in 1956 and Gabriel to abelian categories in 1962, where Krull–Schmidt category now means a category in which idempotents split and objects have local endomorphism rings.

Failure Modes and Common Mistakes

Do not assume a module of infinite length over an artinian ring has a Krull–Schmidt decomposition; $(19.23)$ requires finite generation.
Do not read the ACC-or-DCC hypothesis of $(19.20)$ into $(19.17)$ : $(19.18)$ records that one chain condition is not enough. Over $ℤ$ the module $ℤ$ is indecomposable with the ACC and $End (ℤ) ≅ ℤ$ is not local; the Prüfer group $ℤ (p^{\infty})$ has the DCC and its endomorphism ring, the $p$ -adic integers, is local but has non-nilpotent maximal ideal.
Do not conclude from $t \cdot M ≅ t \cdot N$ that $M ≅ N$ outside a setting where uniqueness holds; cancellation is a corollary of uniqueness, not a general fact.
Do not confuse the multiset of composition factors (Jordan–Hölder) with the multiset of indecomposable summands (Krull–Schmidt). They agree only for semisimple modules.

Best Practices

Quote the version you actually use: $(19.21)$ for hand-checked local endomorphism rings, $(19.22)$ for finite length, $(19.23)$ for finitely generated modules over artinian rings.
When claiming uniqueness, name the reason the endomorphism rings are local — usually finite length via $(19.17)$ .
Record the multiplicities, not just the set of types; the multiplicities are the part that supports cancellation arguments.
If a module fails to have finite length, look first for a class-group-style obstruction before assuming the decomposition is unique.

Historical Notes and Lessons Learned

1909WedderburnUniqueness of decompositions established for finite groups, in the setting that would become the group-theoretic Krull–Schmidt theorem.
1911–1913Remak and SchmidtRemak and then Schmidt sharpen the group-theoretic statement; the result is still called Remak–Krull–Schmidt in group theory.
1925KrullKrull moves the theorem into the setting of operator groups and modules with chain conditions.
1950AzumayaAzumaya isolates the real hypothesis — local endomorphism rings — and proves the version stated here, freeing the theorem from chain conditions on the ambient module.
1956–1962Atiyah, then GabrielAtiyah proves a Krull–Schmidt theorem for coherent sheaves; Gabriel formulates it for abelian categories, where it becomes a definition rather than a theorem.
1960s onwardsSharp failuresSwan's example over a noetherian local domain, and later work of Facchini and others on modules with non-local endomorphism rings, map exactly how much uniqueness survives.

The methodological lesson mirrors that of the Jacobson radical. The classical statements were tied to chain conditions on the module; Azumaya's reformulation moves the hypothesis onto the endomorphism rings of the pieces, where it is both weaker and more clearly the real cause. Conditions stated on the automorphisms of the atoms generalise; conditions stated on the size of the whole do not.

Quick Reference

ExistenceACC or DCC on submodules

(19.20)

UniquenessOne family strongly indecomposable

(19.21)

Finite lengthBoth halves hold

(19.22)

Artinian ringBoth halves for f.g. modules

(19.23)

Key step

id = \sum_{j} (α_{1} β_{j}) |_{M_{1}}

has a unit summand

Cancellation

t \cdot M ≅ t \cdot N \Rightarrow M ≅ N

over artinian

R

Failure

𝔄 \oplus 𝔄 ≅ R \oplus R

R = ℤ [\sqrt{- 5}]

RepairComplete the base ring; see §21

Numbered results used on this page
Reference	Statement	Hypotheses
(19.17)	$End (M)$ is local	$M$ indecomposable of finite length
(19.20)	$M$ is a finite sum of indecomposables	ACC or DCC on submodules of $M$
(19.21)	$r = s$ and $M_{i} ≅ N_{π (i)}$	$N_{j}$ indecomposable, $M_{i}$ strongly so
(19.22)	Existence and uniqueness	$M$ of finite composition length
(19.23)	Existence and uniqueness	$M$ f.g., $R$ right artinian
(19.25)	$M^{K} ≅ N^{K} \Rightarrow M ≅ N$	$R$ f.d. over $k$ , $\dim_{k} M, \dim_{k} N < \infty$

Frequently Asked Questions

Does Krull–Schmidt need both chain conditions?

Existence needs only one: the ACC or the DCC alone suffices for $(19.20)$ . Uniqueness as packaged in $(19.22)$ needs finite length, i.e. both — because it is proved by way of $(19.17)$ , and that result genuinely requires both. What uniqueness really needs is local endomorphism rings, which finite length happens to guarantee.

Why must the summands be strongly indecomposable and not merely indecomposable?

The proof writes ${id}_{M_{1}}$ as a sum of $s$ endomorphisms of $M_{1}$ and needs one of them to be invertible. That inference is exactly the defining property $(19.1) (5')$ of a local ring. An indecomposable module only guarantees that $End (M_{1})$ has no nontrivial idempotents, which does not force any summand of a sum to be a unit.

Where does uniqueness first fail?

Over any Dedekind domain with nontrivial class group. If $[𝔄]$ has order $2$ then $𝔄 \oplus 𝔄 ≅ R \oplus 𝔄^{2} ≅ R \oplus R$ , and $𝔄 not ≅ R$ . Swan's example pushes the failure into commutative noetherian local domains, so being local is no protection.

Is Krull–Schmidt the same as Jordan–Hölder?

No. Jordan–Hölder is about composition factors of a filtration and holds for every module of finite length over every ring. Krull–Schmidt is about indecomposable direct summands. A module can have a unique list of composition factors while its decomposition into indecomposables is not unique, and the two lists coincide only when the module is semisimple.

Does the theorem extend to infinite direct sums?

There is a genuine infinite version — modules with local endomorphism rings satisfy strong exchange properties, and the Crawley–Jónsson–Warfield theory develops this — but it is a separate theorem. The finite induction in the proof above does not extend on its own.

What does the theorem give me computationally?

It makes the output of a decomposition algorithm an invariant. Once you know a module over a finite-dimensional algebra decomposes as $2 U_{1} \oplus 3 U_{2}$ , no other decomposition into indecomposables exists, so the pair of multiplicities is a valid fingerprint for isomorphism testing.

References

T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131, Springer-Verlag, 1991, §19 (pp. 294–310), especially (19.20)–(19.23) and (19.25).
G. Azumaya, “Corrections and supplementaries to my paper concerning Krull–Remak–Schmidt's theorem”, Nagoya Mathematical Journal 1 (1950), 117–124.
F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, 2nd edition, Graduate Texts in Mathematics 13, Springer-Verlag, 1992 (decompositions of modules and the exchange property).
A. Facchini, Module Theory: Endomorphism Rings and Direct Sum Decompositions in Some Classes of Modules, Progress in Mathematics 167, Birkhäuser, 1998.
C. W. Curtis and I. Reiner, Methods of Representation Theory, Volume I, Wiley, 1981 (Krull–Schmidt, principal indecomposables and the Noether–Deuring theorem).
M. F. Atiyah, “On the Krull–Schmidt theorem with application to sheaves”, Bulletin de la Société Mathématique de France 84 (1956), 307–317.

AI Suggested Questions

Write out Azumaya's proof for the case $r = 2$ , $s = 3$ and check each exchange step explicitly.
What exactly is the exchange property, and which modules besides those with local endomorphism rings have it?
Reconstruct Swan's example of non-uniqueness over a commutative noetherian local domain in full detail.
How does completing the base ring restore uniqueness, and where does idempotent lifting enter?
State and prove the Krull–Schmidt theorem for coherent sheaves as Atiyah did.
For which finite-dimensional algebras is the number of indecomposable modules finite, and how does Krull–Schmidt interact with representation type?
Give an example of two non-isomorphic modules over an artinian ring with the same composition factors but different indecomposable decompositions.

01Executive Summary

02Overview

03Learning Objectives

04Definitions

05Core Concepts

Existence is cheap

Uniqueness is expensive

Why finite length is the right sufficient condition

06Key Results

07Proof Techniques and Method

Turn decompositions into idempotents

Make a unit appear in a sum

Split mono into indecomposable is iso

08Worked Example

Two genuinely different decompositions of R2

A case where luck substitutes for the theorem

09Process and Workflow

10Comparison and Classification

11Relationship Map

12Applications and Industry Use

Principal indecomposables and blocks

Noether–Deuring

Canonical form of a module

Vector bundles and projective modules

13Failure Modes and Common Mistakes

14Best Practices

15Historical Notes and Lessons Learned

16Quick Reference

17Frequently Asked Questions

18Related KEVOS Topics

19References

20AI Suggested Questions

Continue learning

Executive Summary

Overview

Learning Objectives

Definitions

Core Concepts

Key Results

Proof Techniques and Method

Worked Example

Two genuinely different decompositions of $R^{2}$

Process and Workflow

Comparison and Classification

Relationship Map

Applications and Industry Use

Failure Modes and Common Mistakes

Best Practices

Historical Notes and Lessons Learned

Quick Reference

Frequently Asked Questions

Related KEVOS Topics

References

AI Suggested Questions